Local behaviour of the second order derivatives of solutions to $p$-Laplace equations

TL;DR

This paper investigates the behavior of second derivatives of solutions to the p-Laplace equation, providing estimates as p approaches 2, which is crucial for understanding regularity in nonlinear PDEs.

Contribution

It offers new $L^{ ext{infinity}}$-type estimates for second derivatives of solutions near p=2, advancing regularity theory for p-Laplace equations.

Findings

Derived $L^{ ext{infinity}}$ estimates for second derivatives as p approaches 2

Enhanced understanding of solution regularity in nonlinear PDEs

Provided bounds that could inform numerical methods for p-Laplace equations

Abstract

We consider the equation $- \Delta_{p} u = f(x)$ in $\Omega,$ where $\Delta_{p}$ is the $p$-Laplace operator. We provide $L^{\infty}$-type estimates for the second derivatives of solutions when $p$ approaches to $2$.